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Abstract. The phenomenon of bubble formation and its detachment from the liquid-vapor interface is highly affected by the application of electric field due to the ...
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ScienceDirect Procedia IUTAM 15 (2015) 86 – 94

IUTAM Symposium on Multiphase flows with phase change: challenges and opportunities, Hyderabad, India (December 08 – December 11, 2014)

Bubble formation in film boiling including electrohydrodynamic forces Vinod Pandeya , Amaresh Dalala , Gautam Biswasa,∗ a Department

of Mechanical Engineering, Indian Institute of Technology, Guwahati 781039, India

Abstract The phenomenon of bubble formation and its detachment from the liquid-vapor interface is highly affected by the application of electric field due to the electrohydrodynamic (EHD) forces. A numerical study is performed using a CLSVOF (Combined levelset and volume of fluid) solver for interface tracking, to analyse the changes in nature of bubble growth in film boiling with the changes in degree of superheat and with the electrohydrodynamic forces. The instability behaviour at the interface changes from Rayleigh-Taylor to Taylor-Helmholtz as the degree of superheat is increased. EHD forces results in the increase in the bubble formation sites due to the decrease in the critical wavelength. c 2015  2014The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Indian Institute of Technology, Hyderabad. Peer-review under responsibility of Indian Institute of Technology, Hyderabad. Keywords: Electrohydrodynamics; Bubble formation; Film boiling; Interface capturing; CLSVOF ;

1. Introduction Boiling heat transfer is highly influenced by the growth and detachment of the bubbles at the liquid-vapor interface. This intricate phenomenon is itself dependent on the instabilities at the interface. The important parameters affecting the interface dynamics are the various forces acting at the interface. These forces can have stabilizing or destabilizing nature on the growth of the disturbances. The effects of surface tension, gravitational acceleration and electric field with heat flux, dominate the flow behaviour in film boiling. Bubble formation in film boiling is the result of the growth of disturbances at the liquid-vapor interface with the transfer of latent heat from vapor phase to liquid phase. As a result, bubbles evolve alternatively from the crest and core of the most critical wavelength of disturbances called as nodes and antinodes, respectively. This phenomenon was first observed numerically by Son and Dhir 1 using level-set approach. For increasing wall superheat, the changes in the bubble release pattern was observed. The morphological changes in bubble formation with wall superheat in case of horizontal film boiling was again studied by Tomar et al. 2 for larger domain. The spacing between the bubble formation sites was found to be correctly matching with the analytical relation of Berenson 3 for lower wall superheat and with that of the Panzarella 4 for higher wall superheat value. Similar phenomena was observed by Hens et al. 5 who observed the deviation of instability nature at the ∗

Corresponding author. Tel.: +91-361-269-0401 ; fax: +91-361-269-2321. E-mail address: [email protected]

2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of Indian Institute of Technology, Hyderabad. doi:10.1016/j.piutam.2015.04.013

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interface from Rayleigh-Taylor to Taylor-Helmholtz. The nature of boiling regime was observed to change from laminar at lower superheat to turbulent behaviour at higher superheat where the periodicity of detachment of bubbles vanishes. With further increase of superheat plug or slug flow regime was also observed. Electrohydrodynamics (EHD) in boiling has been an extensive area of research for many authors for the various advantages of the effects of electric force on the boiling characteristics. In the problems involving interfacial regions coupled with electric field, surface interactions play a dominant role as explained by Melcher and Taylor 6 . One of the significant advantages of application of electric field is in the low gravity application as studied by Di Marco and Grassi 7 . The electrical forces are found to dominate in low gravity boiling and thus restore the same value of critical heat flux as in normal gravity on terrestrial. Lovenguth and Hanesian 8 through their experiments showed a significant increase in the peak heat flux value on application of non-uniform electric field on boiling heat transfer mechanism. Ogata and Yabe 9 found significant enhancement in heat transfer rate in their experiment for boiling in a mixture of conducting and dielectric fluids. Similarly, electric field effect on nucleate boiling with subcooling and heat-up rate was studied by Carrica et al. 10 . Cho et al. 11 studied the effect of a uniform electric field on an insulating bubble attached to a wall. The effect of electric field strength on contact angle and contact radius was determined with one parameter kept constant and other varying. On somewhat different ground, the droplet dynamics (interaction, deformation, breakup, equilibrium shape etc.) in the presence of an electric field were studied separately by Harris and Basaran 12 , Baygents et al. 13 , Ha and Yang 14 and Notz and Basaran 15 . Elongation of vapor bubbles in the presence of electric field was observed by Verplaetsen and Berghmans 16 in their study of heat transfer rate during film boiling in a stagnant fluid on a horizontal surface. Similar observation was experienced by Cheng and Chaddock 17 in a uniform electric field during nucleate boiling. Larger surface area to volume ratio of bubbles formed signifies faster growth due to increased heat transfer. Welch and Biswas 18 simulated the film boiling including electrohydrodynamic forces incorporating the mass transfer model and surface tension model in the CLSVOF method. The medium was perfect dielectric, viscous, heat conducting and incompressible. The increase in bubble release frequency with a decrease in critical wavelength was observed. Tomar et al. 19 presented a new approach of representing the electric field surface force in their volume of fluid (VOF) based methodology. They also presented an efficient means of calculation of electrical properties in the transition region. The simulation was performed using both dielectric-dielectric and conducting-conducting fluid interfaces to analyze the effect of surface forces and the transition region. Tomar et al. 20 using their formulation simulated the boiling phenomenon for a larger domain size to accommodate for multiple bubble formation sites. Simulations were performed for lower and higher superheat values and increase in bubble density was observed for a given domain with the increase in electric field intensity. Application of electric field in boiling has found to result in a significant enhancement in heat transfer characteristics. An advantage of utilising an EHD technique is that the heat transfer performance can easily be controlled by varying the applying voltage. Some of the observed and reported effects of electric field on the boiling phenomenon are • • • • • • •

Enhancement in heat and mass transfer across the interface. Enhanced interfacial motion. Decrease in the number of active nucleated sites during nucleate boiling. In sufficiently strong electric field, absence of transition region between nucleate and film boiling. Increase in critical heat flux and minimum heat flux for film boiling. Shorter bubble separation distance and higher bubble release frequency. Increase in space averaged Nusselt number.

The coupling of electric field in film boiling mechanism leads to various modification in the physical behaviour of flow. The influence of electric field on film boiling can be inferred from the additional destabilising term introduced by Johnson 21 in the dispersion relation (Eq. 1) by Zuber 22 which can be written with electric field term as, ω2 =

σ (ρl − ρv ) f (El , l , v ) 2 m3 − gm − m ρl + ρv ρl + ρv ρl + ρv

(1)

where the subscripts l and v represent the liquid and vapor phase, respectively. σ, ρ and  are the surface tension coefficient, density and the relative dielectric permittivity of the medium, respectively, g is the acceleration due to

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gravity, m is the wave number and El is the magnitude of electric field intensity. The function f (El , l , v ) is defined as, l (l − v )2 v (l − v )2 f (E, l , v ) = (2) 0 El2 = 0 Ev2 v (l + v ) l (l + v ) where Ev = l El /v is the magnitude of electric field intensity in the vapor phase. It can be observed from Eq. (1) that like gravity, the electric field also has destabilising effect on the interface. Johnson 21 also derived an expression for the most dominant wavelength λ J under the influence of electric field as, λJ =

6πσ    v )gσ f 1 + 1 + 3(ρl −ρ f2

(3)

Di Marco and Grassi 7 introduced an important dimensionless number El∗ called as electrical influence number on the basis of which they predicted the most dominant wavelength as, √ 3 λE = λ B (4) √ ∗ El + El∗2 + 3 where,

f (El , l , v ) El∗ =  (ρl − ρv )σg

and λB is the Berenson’s 3 most dominant wavelength without the electric field given by,  3σ λB = 2π (ρl − ρv )g

(5)

(6)

In the present work, we follow the CLSVOF alogrithm used by Welch and Biswas 18 and is proved to be of comparatively better numerical ease and accuracy than other existing methods by Gerlach et al. 23 . The implementation of EHD forces in the numerical approach follows the expression given by Tomar et al. 19 . Numerical simulations were performed to study the effects of electric field on the film boiling with different wall superheat. The pressure equation and the electric field equations are solved using HYPRE-multigrid solver. 2. Numerical Formulation The two-phase system considered here consists of two incompressible homogeneous fluids with a moving interface. In saturated film boiling, the incessant phenomena of bubble growth at the interface occurs in a very minor time and length scale which thereby, can be understood effectively through numerical methodology. Simulating the flow behaviour in flows incorporating sharp interfaces requires a physical understanding of the forces acting at the interface and then modelling the flow in a proper mathematical form. The interface is considered to be as a free boundary where the flow variables exhibit discontinuities. 2.1. Governing equations A set of equations can be written for an incompressible Newtonian fluid consisting of the mass, momentum and energy conservation equations as: ∇·U=0

(7)

ρ [Ut + ∇ · (UU)] = −∇p + ρg + ∇ · (2μDv ) + f sv

(8)

∂T kv 2 + U · ∇T = ∇T ∂t ρc p

(9)

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where U = (u, v) is the velocity vector, t is the time, p is the pressure, g = (0, −g) is the gravitational acceleration, Dv = 12 (∇U) + (∇U)T is the rate of deformation tensor and ρ and μ are the density and viscosity based on the volume fraction function F as ρ (F) = ρl F + ρv (1 − F) (10) μ (F) = μl F + μv (1 − F) The volume fraction function Fi, j is defined on each cell (i, j) as the fraction of the liquid inside a cell. ⎧ ⎪ ⎪ 0 if gaseous cell ⎪ ⎪ ⎪ ⎨ F=⎪ if liquid cell 1 ⎪ ⎪ ⎪ ⎪ ⎩0 < F < 1 if mixed cell

(11)

(12)

The last term f sv in momentum conservation equation (Eq. 8) is the surface tension force per unit volume defined by the continuum surface force model of Brackbill et al. 24 as ˆ s f sv = σκnδ

(13)

where σ is the surface tension coefficient, κ is the mean curvature of the interface, nˆ is the unit normal vector of the interface and δ s is the interface delta function. In the energy conservation equation (Eq. 9), T , kv and c p are the temperature, thermal conductivity and the specific heat capacity of the vapor phase, respectively. For the saturated film boiling case, energy conservation equation is solved only for the vapor phase as the liquid phase is considered to be at constant saturation temperature. For the flow coupled with electric field, some additional electrical laws have to be taken care of. The assumptions include the negligible induced magnetic field due to the small dynamic current present and the absence of any external magnetic field. Thus, irrotationality of the electric field is considered,i.e, ∇×E=0

(14)

Also, in the present study, the medium is considered as a dielectric with no free charge present. For this case, the electric displacement vector D = 0 E according to Gauss law is given by ∇·D=0

(15)

This implies electric field to be a gradient of some scalar function as E = −∇ψ

(16)

∇ · (0 ∇ψ) = 0

(17)

where ψ is the electric potential function. Hence,

At the interface, the continuity of electric potential and the normal component of electric field satisfies the jump condition as, 0 ∇ψ · n = 0 (18) An additional volume force term is added in the momentum equation (Eq.8), given by, 1 fvE = ∇ · τE = − E 2 ∇0  2

(19)

where τE is the Maxwell stress tensor defined as,

 1 2 τ = 0 EE − E I 2 E

where I is the identity tensor.

(20)

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2.2. CLSVOF approach The CLSVOF formulation 23,2 applied here utilizes a smoothed Heaviside function H(φ) defined as, ⎧ ⎪ ⎪ 0 if φ < −δ ⎪ ⎪ ⎪ ⎨ 1 φ 1   πφ  Hδ (φ) = ⎪ if |φ| ≤ δ ⎪ 2 + 2δ + 2π sin δ ⎪ ⎪ ⎪ ⎩1 if φ > δ

(21)

where φ is the smooth level-set function. This H(φ) is considered as the smoothed void fraction function F˜ to be used in calculating density and viscosity in the transition region through Equations (10) and (11), respectively. The normal vector can be calculated following Sussman et al. 25 as n = ∇Hδ (φ) The curvature κ is then given by κ = −∇ · nˆ

where nˆ = ∇ ·

(22) ∇n |∇n|

(23)

2.3. Boundary conditions The boundary conditions on the side walls are symmetry boundary conditions, i.e. ∂T ∂F ∂φ ∂ψ ∂v = 0, = 0, = 0, = 0, =0 ∂x ∂x ∂x ∂x ∂x where L is the right side boundary distance, depending on the simulation performed. At the top wall, outflow boundary conditions are used, i.e. at x = 0 and x = L : u = 0,

at y = H :

∂u ∂v ∂T ∂F ∂φ = = = = = 0; P = P0 ∂y ∂y ∂y ∂y ∂y

The outlet pressure is taken as the saturation pressure less than the hydrostatic pressure difference from the initial film level to the outlet. The bottom wall is at a constant temperature, i.e. at y = 0 : T = T sat + ΔT sup

(24)

Electric field is applied with a specified electric field strength at the top wall and zero electric potential at the bottom wall. At the liquid-vapor interface, the temperature (T I ) is assumed to be equal to the saturation temperature at the liquid pressure, i.e., T I = T sat (25) The jump in normal velocity at the interface due to the phase change can be given as, 

1 1 q · nˆ (U − Ul ) · nˆ = − ρl ρv hlv

(26)

The kinetic energy term, viscous work term and the viscous dissipation are neglected in the energy jump condition. The force balance at the interface considering the pressure difference across interface, viscous stresses, surface tension force and electric stress forces can be given as, p1 − p2 = σκ + n · (τv · n) + fsE,d · n

(27)

where pi represents the pressure in medium i = 1, 2. τv is the viscous stress tensor and fsE is the surface force due to the applied electric field which for perfectly dielectric fluids acts normal to the interface and is given by fsE ·n = n·(τE · n).

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Fig. 1. (a) Interface evolution with time for smaller wall superheat of 2 K.; (b) Interface evolution with time for larger wall superheat of 18 K. Table 1. Properties for liquid and vapor phases of water at near critical pressure (Pr = 0.99)

Water near critical condition: T sat = 646 K; P sat = 21.9 MPa; hlv = 276.4 kJ/kg;σ = 0.07 mN/m Phase

Density (ρ) (kg/m3 )

Viscosity (μ) (μN s/m2 )

Conductivity (k) (W/mK)

Specific heat (c p ) (kJ/kgK)

Relative permittivity ()

Liquid Vapor

402.4 242.7

46.7 32.38

0.5454 0.5383

2.18 ×102 3.52 ×102

7.35 3.71

3. Results and discussions To verify the changes in evolution of liquid-vapor interface with the varying wall superheat, the simulations have been performed for lower and higher superheat for 5λB domain length for the saturated water at near critical condition. The properties of water at that condition have been specified in Table 1. The boundary conditions taken is as mentioned in Section 2.3. A uniform grid is used with size Δx = Δy = λB /252 for all cases with a constant time step of Δt = 5 × 10−6 s. The number of bubbles obtained and their evolution with time for lower superheat (2 K) and for the higher superheat (18 K) can be observed from Fig. 1. As observed by Hens et al. 5 , at lower superheat six bubbles in 5λB domain refers to the Rayleigh-Taylor instability and five bubbles in the same domain refers to the TaylorHelmholtz instability behaviour at the interface. Thus, at lower superheat, lubrication approximation dominates as the prominence of viscosity increases the number of bubble formation sites. Whereas, at higher superheat the inviscid approximation can be inferred. To determine the effect of electric field on the morphology of bubbles emanating from the interface, we simulated the boiling of the same liquid as in previous simulation but with higher wall superheat,i.e. 40 K, to account for the dielectric behaviour of the fluid. This simulation is performed to validate the results from those obtained by Tomar et al. 20 who obtained eleven and a half bubbles in 3λB domain. The evolution of bubbles with time have been shown in Fig. 2 and Fig. 3 for the boiling with applied electric field and without electric field, respectively. Uniform electric field of strength 2 × 105 V/m have been applied in the vertically upward direction. As can be observed from the figure that the number of bubble formation sites increase enormously due to the destabilising EHD forces favouring the bubble growth at the interface.

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Fig. 2. Morphological evolution of liquid-vapor interface with applied uniform electric field strength of 2 × 105 V/m and wall superheat of 40 K.

Fig. 3. Morphological evolution of liquid-vapor interface with wall superheat of 40 K.

To elucidate the effect of electric force on interface, comparison is shown using Fig.2 and Fig.3 for the growth of bubbles with applied electric field and without electric field. As expected, the separation between the bubbles is almost same as the Berenson’s wavelength, with three bubbles formed in 3λB domain. Whereas, for the applied electric field the number of bubbles emanating is equal to twelve signifying the increase in the destabilising effect at the interface. The deformation of bubbles prolately in the direction of applied electric field have already been reported by many authors. This deformation decreases the bubble size and reduces the buoyancy forces during the bubble growth, leading to higher necking distance from the heating surface. Further, we performed simulation for the wall superheat of 5 K with varying magnitudes of electric field strength to verify the destabilizing nature of EHD forces. The domain size is taken as 5λB ×λB with the other parameters unaltered except the electric field strength value at the top wall. The values of electric field strength applied are 1 × 105 V/m and 2 × 105 V/m and the results obtained are as shown in Fig. 4 and Fig. 5, respectively. As can be observed from the figures that the bubble spacing decreases significantly with the increase in the value of applied electric field. For 1 × 105 V/m, number of bubbles obtained is nine whereas for 2 × 105 V/m, the number of bubbles obtained increases to nineteen. We also observed the effect of wall superheat on the bubble detachment distance from the wall. For boiling with 5 K wall superheat, although the number of bubble formation sites increase significantly but the detachment of bubbles from the vapor film is observed to be at much less distance from the wall as compared to the boiling with 40 K wall superheat. Thus, the effect of the heat flux on the morphological changes in bubbles can be observed to be significant. This phenomenon can also be observed in boiling without applied electric field from the Fig. 1.

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Fig. 4. Evolution of liquid-vapor interface for the wall superheat of 5 K and applied uniform electric field strength of 1 × 105 V/m.

Fig. 5. Evolution of liquid-vapor interface for the wall superheat of 5 K and applied uniform electric field strength of 2 × 105 V/m.

4. Conclusion On the basis of the simulations performed, an illustrative explanation of morphogical evolution of the liquid-vapor interface has been performed for the boiling with the applied electric field and without the electric field. The increase in critical wavelength for varying wall superheat is justified through the simulation performed for low wall superheat and higher wall superheat. A significant increase in bubble density is observed on application of electric field for both higher wall superheat and lower wall superheat. This signifies the increase in destabilsing nature of the electric field on the disturbances at the liquid-vapor interface. An increase in bubble detachment distance from the heated wall is observed for the higher wall superheat as compared to the lower wall superheat for boiling with electric field and without electric field, both. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

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